26 Physical Basis of Spatial Distortions in Magnetic Resonance Images

Peter Jezzard 425

27 Physical and Biological Bases of Spatial Distortions in Positron Emission Tomography Images Magnus Dahlbom and Sung-Cheng (Henry Huang) 439

28 Biological Underpinnings of Anatomic Consistency and Variability in the

Human Brain N Tzourio-Mazoyer, F. Crivello, M. Joliot, and B. Mazoyer 449

29 Spatial Transformation Models Roger P. Woods 465

30 Validation of Registration Accuracy Roger P. Woods 491

31 Landmark-Based Registration Using Features Identified Through Differential Geometry Xavier Pennec, Nicholas Ayache, and Jean-Philippe Thirion 499

32 Image Registration Using Chamfer Matching Marcel Van Herk 515

33 Within-Modality Registration Using Intensity-Based Cost Functions

Roger P. Woods 529

34 Across-Modality Registration Using Intensity-Based Cost Functions

Derek L.G. Hill and David J. Hawkes 537

35 Talairach Space as a Tool for Intersubject Standardization in the Brain

Jack L. Lancaster and Peter T. Fox 555

36 Warping Strategies for Intersubject Registration Paul M. Thompson and Arthur W. Toga 569

37 Optimizing the Resampling of Registered Images William F. Eddy and Terence K. Young 603

38 Clinical Applications of Image Registration Robert Knowlton 613

39 Registration for Image-Guided Surgery Eric Grimson and Ron Kikinis 623

40 Image Registration and the Construction of Multidimensional Brain Atlases

Arthur W. Toga and Paul M. Thompson 635

Roger P. Woods

UCLA School of Medicine

The goal of image registration is to determine a spatial transformation that will bring homologous points in images being registered into correspondence. In the simplest cases, the mathematical form of the desired spatial transformation can be limited by simple physical principles. For example, when registering images acquired from the same subject, it is often possible to assume that the body part being imaged can be treated as a rigid body, which leads to a highly constrained spatial transformation model. Unfortunately, physical processes involved in the acquisition and reconstruction of medical images can cause artifacts and lead to violations of the rigid body model, even when the object being imaged adheres strictly to rigid body constraints. Potential sources of such distortions are prevalent in magnetic resonance (MR) and positron emission tomography (PET) images. So far as is practical, these distortions should be corrected explicitly using methods that estimate the appropriate correction parameters independent of the registration process itself, since this will improve both the speed and the accuracy with which rigid body movements can be estimated. The chapters "Physical Basis of Spatial Distortions in Magnetic Resonance Images" and "Physical and Biological Bases of Spatial Distortions in Positron Emission Tomography Images" describe the physical processes that lead to distortions in these common imaging modalities. Distortions of soft tissues can also lead to nonlinear effects that violate rigid body assumptions, a topic addressed in the chapters "Physical and Biological Bases of Spatial Distortions in Positron Emission Tomography Images" and "Image Registration Using Chamfer Matching." Such distortions are governed by complex properties such as tissue elasticity that are much more difficult to model than the physical factors associated with image acquisition or reconstruction. Registration of images acquired from different subjects represents the extreme end of the spectrum, where developmental factors including genetics, environment, and random influences all contribute to the complex differences between subjects. The chapter "Biological Underpinnings of Anatomic Consistency and Variability in the Human Brain" provides an overview of the complexity of this most difficult registration problem in the context of the human brain.

Much of the work that has been done on image registration to date has concerned itself with spatial transformations that are subject to linear constraints. The rigid body model is the set of linear constraints most commonly utilized, but more relaxed linear models are also well suited for dealing with certain types of image distortions such as errors in distance calibration. Even in the context of intersubject registration, where highly nonlinear transformations would be required for perfect registration, linear transformations can provide useful approximations. The mathematical and geometric properties of linear spatial transformations are discussed in detail in the chapter "Spatial Transformation Models." One of the key attributes of linear models is that only a small amount of information is required to define a spatial transformation. For example, the identification of three point landmarks in each of two different tomographic image data sets is sufficient to estimate the three-dimensional rigid body transformation needed to register the two sets of images with reasonable accuracy. Medical images commonly contain far more spatial information than this minimal requirement, and this redundancy can be exploited to achieve highly accurate registration results with errors often smaller than the size of a voxel. The redundancy also provides a mechanism whereby registration accuracy can be objectively evaluated. As the appropriate spatial transformation model becomes less constrained for example, in the case of intersubject registration the redundancy is reduced or even eliminated entirely. Validation becomes much more complicated when the mathematical form of the spatial transformation model is nonlinear and entails many degrees of freedom. Various strategies for evaluating registration accuracy are discussed in the chapter "Validation of Registration Accuracy."

Another consequence of the redundancy of spatial information for deriving linear spatial transformations is the fact that diverse approaches can be successfully used for registration. Historically, identification of point landmarks has been the most straightforward strategy employed. Most commonly, human intervention has provided the anatomic expertise needed to identify homologous structures in the images to be registered, and the mathematics for converting a set of point landmarks into an optimal spatial transformation is straightforward. More recent work with point landmarks has focused on eliminating the need for human intervention by identifying unique features within the data sets and defining homologies among these features using computerized methods. Work in this area is reviewed in the chapter "Landmark-Based Registration Using Features Identified Through Differential Geometry." This novel approach to landmark identification produces a much higher degree of redundancy of spatial information than can be practically achieved by a human observer, and has produced marked improvements in accuracy over those routinely achieved using manually identified landmarks. As a method for extracting spatial information, an alternative to the explicit identification of point landmarks is the identification of curves or surfaces. Although they lack explicit one-to-one correspondences of landmark points, surface matching algorithms are nonetheless able to minimize the distances between corresponding curves or surfaces to achieve accurate registration. The distance between surfaces at each point varies in a complex manner as the parameters of the spatial transformation model are varied, and efficient strategies for computing these distances have a substantial impact on performance of such algorithms. Chamfer matching represents one approach to streamlining the computation of distances, and variations of this strategy are discussed in the chapter "Image Registration Using Chamfer Matching." The use of contours for registration can be viewed as being a somewhat more abstract strategy than point landmarks for tapping into the spatial information contained within images.

Registration methods that are based on image intensities represent an even more abstract strategy. Instead of minimizing a real-world distance that has an obvious and intuitive link to the notion of an optimal set of registration parameters, intensity-based methods substitute a "cost function" that reflects similarities in image intensities. Since no anatomic features are explicitly identified, intensity-based methods must include some intrinsic model of how various intensities in one image should correspond to intensities in the other image. For registration of images acquired with the same imaging modality, the selection and optimization of a suitable cost function is fairly straightforward, as discussed in the chapter "Within-Modality Registration Using Intensity-Based Cost Functions." When the problem is generalized to include registration of images from different modalities, more sophisticated cost functions and minimization strategies are needed, as discussed in the chapter "Across-Modality Registration Using Intensity-Based Cost Functions."

Intersubject registration warrants separate consideration because of the complex nature of this problem. Current work in this area is largely restricted to the brain, reflecting the tremendous interest in the relationships between structure and function in this complex organ. Unlike other organs where function is not highly differentiated, brain regions separated by small distances often have highly distinct functions. Consequently, improvements in methods for registering homologous regions have important implications for research and for clinical applications. Linear spatial transformation models provided much of the initial framework for research in this area, and the notion of "Talairach space," which was originally defined in terms of linear transformations, remains an important concept in brain research. The chapter "Talairach Space as a Tool for Intersubject Standardization in the Brain" reviews the origins and modern applications of this particular frame of reference for describing brain locations. Currently research on intersubject registration in the brain is focused on the use of nonlinear warping strategies, and an overview of many of the diverse methods under investigation is provided in the chapter "Warping Strategies for Intersubject Registration."

In many instances, the primary focus of image registration is to quantify movements so that their influence on the data can be minimized or even eliminated. The registration process is essentially a process for removing the effects of an unwanted confounding movement from the data. Once the desired spatial transformation has been derived, image resampling and interpolation models must be utilized to compensate for the movement and create registered images. Although image interpolation can be viewed as an issue in image quantification (see the Quantification section, "Image Interpolation and Resampling"), certain unique issues arise only in the context of image registration. These issues are addressed in the chapter "Optimizing the Resampling of Registered Images."

Advances in registration methods are closely linked to advances in clinical and research applications. In many ways, the advances are reciprocal, with improved imaging methods leading to improved registration techniques, which in turn lead to the recognition of new clinical and research applications for the methods. This in turn leads to increased demand for registration methods that are even more accurate and efficient. The diversity of registration problems to be solved, together with the performance requirements imposed by diverse clinical and research contexts, accounts in part for the large number of registration strategies that have been published in the medical literature. This reciprocal relationship is likely to persist in the future, and the final three chapters in this section, "Clinical Applications of Image Registration," "Registration for Image-Guided Surgery," and "Image Registration and the Construction of Multidimensional Brain Atlases," provide an overview of the types of problems currently being addressed by image registration techniques, as well as problems that will need to be addressed through image registration in the future.

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